Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ \frac{x}{2^{2}-1}+\frac{x^{2}}{3^{2}-1}+\frac{x^{3}}{4^{2}-1}+\frac{x^{4}}{5^{2}-1}+\cdots $$

Step-by-Step Solution

Answer

The convergence set is \((-1, 1)\).

Step 1: Identify the General Term

The given series can be written in the form of a power series \( \sum_{n=1}^{\infty} a_n x^n \). By inspecting the series, we observe that the denominators are consecutive squares minus one. The general term \( a_n \) is then \( a_n = \frac{1}{((n+1)^2 - 1)} = \frac{1}{n^2 + 2n} \). Thus, the series is \( \sum_{n=1}^{\infty} \frac{x^n}{n^2 + 2n} \).

Step 2: Apply the Ratio Test

The Ratio Test for absolute convergence examines the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| \). For our series, \( a_{n+1} = \frac{1}{(n+2)^2 + 2(n+2)} \), so the ratio is:\[\left| \frac{\frac{x^{n+1}}{(n+2)^2 + 2(n+2)}}{\frac{x^n}{n^2 + 2n}} \right| = \left| x \right| \cdot \frac{n^2 + 2n}{(n+2)^2 + 2(n+2)}.\]

Step 3: Simplify the Expression

Simplify the expression from the Ratio Test:\[\frac{n^2 + 2n}{(n+2)^2 + 4n + 4} = \frac{n^2 + 2n}{n^2 + 4n + 4}.\]As \( n \to \infty \), the highest power terms dominate:\[\lim_{n \to \infty} \frac{n^2 + 2n}{n^2 + 4n + 4} = 1.\]Thus, the limit simplifies to \(|x| \cdot 1 = |x|\).

Step 4: Determine Convergence Conditions

According to the Ratio Test, the series converges if \(|x| < 1\). Therefore, the set of convergence is the interval \((-1, 1)\). This series converges absolutely for \(x\) in this interval.

Key Concepts

Power Series

A power series is a series of the form \[\sum_{n=0}^{\infty} a_n (x-c)^n\] where \(a_n\) are coefficients and \(c\) is the center of the series. In our problem, the series is centered at 0 because it does not involve shifting by any \(c\). Each term in a power series is a power of \(x\), making it essential for solutions in calculus, especially for functions that cannot be expressed using elementary functions.

• **General Structure:** The general form involves coefficients \(a_n\), which multiply terms \(x^n\).
• **Purpose:** Power series are used to approximate complex functions as polynomials, which are easier to manipulate and integrate.

In many math and physics applications, knowing how to express complicated functions as power series can simplify problems significantly. It helps in both theoretical analyses and practical computations, such as evaluating limits or solving differential equations.

Absolute Ratio Test

The Absolute Ratio Test is a method for determining the convergence of a series. It can particularly ascertain the radius of convergence for a power series. In applying the Absolute Ratio Test:
1. Consider the limit: \[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| \]2. Analyze: - If \(L < 1\), the series converges absolutely. - If \(L > 1\), the series diverges. - If \(L = 1\), the test is inconclusive.The Absolute Ratio Test is particularly useful for determining the interval of convergence. By examining the terms \(a_n\), students can predict behavior across various segments of \(x\), identifying the values for which the series converges smoothly.

Convergence Interval

The interval of convergence is the range of values of \(x\) in which a power series converges to a limit. In our exercise, we found that the series converges for values of \(x\) within the interval \((-1, 1)\), determined by evaluating the limit of the ratio test result.

• **Boundaries of the interval:** The series does not converge at the endpoints, \(x = 1\) and \(x = -1\), making it a strict open interval.
• **Importance:** Knowing the convergence interval is important because it tells us over which portion of the number line the series behaves properly, providing accurate approximations and sums.
• **Practical Use:** Within this interval, the series can comfortably be used for numerical calculations or as a functional representation.

Understanding the convergence interval helps students grasp how power series behave and when they can reliably use them for computations.

General Term Identification

Identifying the general term \(a_n\) of a power series is a crucial step in understanding the series' behavior. In our problem, the general term is given by:\[a_n = \frac{1}{n^2 + 2n}\]where \(n+1\) is used to form the pattern seen in denominators of the series terms. The ability to spot and represent this general term correctly allows us to use various convergence tests accurately.

• **Pattern Recognition:** By observing the numerators and denominators, students can determine a consistent sequence in the series.
• **Application:** Once the general term is recognized, it forms the basis for applying convergence tests, such as the Ratio Test.

Through general term identification, students gain insights into the underlying structure of the series, making it easier to determine both the convergence behavior and approximate functions using power series.